Newtonian fluid
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{{Continuum mechanics}}
A '''Newtonian fluid''' (named for [[Isaac Newton]]) is a [[fluid]] whose [[shear stress|stress]] versus rate of [[strain (materials science)|strain]] curve is linear and passes through the [[Origin (mathematics)|origin]]. The constant of proportionality is known as the [[viscosity]].
==Definition==
A simple equation to describe Newtonian fluid behaviour is
:<math>\tau=\mu\frac{du}{dx}</math>
where
:<math>\tau</math> is the shear stress exerted by the fluid ("[[Drag (physics)|drag]]") [Pa]
:<math>\mu</math> is the fluid viscosity - a constant of proportionality [Pa·s]
:<math>\frac{du}{dx}</math> is the velocity gradient perpendicular to the direction of shear [s<sup>−1</sup>]
In common terms, this means the fluid continues to flow, regardless of the forces acting on it. For example, water is Newtonian, because it continues to exemplify fluid properties no matter how fast it is stirred or mixed. Contrast this with a [[non-Newtonian fluid]], in which stirring can leave a "hole" behind (that gradually fills up over time - this behaviour is seen in materials such as pudding, starch in water ([[oobleck]]), or, to a less rigorous extent, sand), or cause the fluid to become thinner, the drop in viscosity causing it to flow more (this is seen in non-drip [[paint]]s, which brush on easily but become more viscous when on walls).
For a Newtonian fluid, the viscosity, by definition, depends only on [[temperature]] and [[pressure]] (and also the chemical composition of the fluid if the fluid is not a pure substance), not on the forces acting upon it.
If the fluid is [[incompressible fluid|incompressible]] and viscosity is constant across the fluid, the equation governing the shear stress, in the [[Cartesian coordinate system]], is
:<math>\tau_{ij}=\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)</math>
with comoving stress tensor <math>\mathbb{P}</math> (also written as <math>\mathbf{\sigma}</math>)
:<math>\mathbb{P}_{ij}= - p \delta_{ij} + \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)</math>
where, by the convention of [[Tensor|tensor]] notation,
:<math>\tau_{ij}</math> is the shear stress on the <math>i^{th}</math> face of a fluid element in the <math>j^{th}</math> direction
:<math>u_i</math> is the velocity in the <math>i^{th}</math> direction
:<math>x_j</math> is the <math>j^{th}</math> direction coordinate
If a fluid does not obey this relation, it is termed a [[non-Newtonian fluid]], of which there are several types, including [[polymer]] solutions, molten polymers, many solid suspensions and most highly viscous fluids.
{{Physics-footer}}
[[Category:viscosity]]
[[Category:Fluid dynamics]]
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